Optimal. Leaf size=246 \[ \frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3810, 2207,
2225} \begin {gather*} \frac {d (c+d x) e^{-6 e-6 f x}}{144 a^3 f^2}-\frac {3 d (c+d x) e^{-4 e-4 f x}}{64 a^3 f^2}+\frac {3 d (c+d x) e^{-2 e-2 f x}}{16 a^3 f^2}+\frac {(c+d x)^2 e^{-6 e-6 f x}}{48 a^3 f}-\frac {3 (c+d x)^2 e^{-4 e-4 f x}}{32 a^3 f}+\frac {3 (c+d x)^2 e^{-2 e-2 f x}}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}+\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2207
Rule 2225
Rule 3810
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+a \coth (e+f x))^3} \, dx &=\int \left (\frac {(c+d x)^2}{8 a^3}-\frac {e^{-6 e-6 f x} (c+d x)^2}{8 a^3}+\frac {3 e^{-4 e-4 f x} (c+d x)^2}{8 a^3}-\frac {3 e^{-2 e-2 f x} (c+d x)^2}{8 a^3}\right ) \, dx\\ &=\frac {(c+d x)^3}{24 a^3 d}-\frac {\int e^{-6 e-6 f x} (c+d x)^2 \, dx}{8 a^3}+\frac {3 \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{8 a^3}-\frac {3 \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{8 a^3}\\ &=\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {d \int e^{-6 e-6 f x} (c+d x) \, dx}{24 a^3 f}+\frac {(3 d) \int e^{-4 e-4 f x} (c+d x) \, dx}{16 a^3 f}-\frac {(3 d) \int e^{-2 e-2 f x} (c+d x) \, dx}{8 a^3 f}\\ &=\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}-\frac {d^2 \int e^{-6 e-6 f x} \, dx}{144 a^3 f^2}+\frac {\left (3 d^2\right ) \int e^{-4 e-4 f x} \, dx}{64 a^3 f^2}-\frac {\left (3 d^2\right ) \int e^{-2 e-2 f x} \, dx}{16 a^3 f^2}\\ &=\frac {d^2 e^{-6 e-6 f x}}{864 a^3 f^3}-\frac {3 d^2 e^{-4 e-4 f x}}{256 a^3 f^3}+\frac {3 d^2 e^{-2 e-2 f x}}{32 a^3 f^3}+\frac {d e^{-6 e-6 f x} (c+d x)}{144 a^3 f^2}-\frac {3 d e^{-4 e-4 f x} (c+d x)}{64 a^3 f^2}+\frac {3 d e^{-2 e-2 f x} (c+d x)}{16 a^3 f^2}+\frac {e^{-6 e-6 f x} (c+d x)^2}{48 a^3 f}-\frac {3 e^{-4 e-4 f x} (c+d x)^2}{32 a^3 f}+\frac {3 e^{-2 e-2 f x} (c+d x)^2}{16 a^3 f}+\frac {(c+d x)^3}{24 a^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.85, size = 371, normalized size = 1.51 \begin {gather*} \frac {\text {csch}^3(e+f x) \left (81 \left (8 c^2 f^2+4 c d f (3+4 f x)+d^2 \left (7+12 f x+8 f^2 x^2\right )\right ) \cosh (e+f x)+8 \left (18 c^2 f^2 (1+6 f x)+6 c d f \left (1+6 f x+18 f^2 x^2\right )+d^2 \left (1+6 f x+18 f^2 x^2+36 f^3 x^3\right )\right ) \cosh (3 (e+f x))+729 d^2 \sinh (e+f x)+1620 c d f \sinh (e+f x)+1944 c^2 f^2 \sinh (e+f x)+1620 d^2 f x \sinh (e+f x)+3888 c d f^2 x \sinh (e+f x)+1944 d^2 f^2 x^2 \sinh (e+f x)-8 d^2 \sinh (3 (e+f x))-48 c d f \sinh (3 (e+f x))-144 c^2 f^2 \sinh (3 (e+f x))-48 d^2 f x \sinh (3 (e+f x))-288 c d f^2 x \sinh (3 (e+f x))+864 c^2 f^3 x \sinh (3 (e+f x))-144 d^2 f^2 x^2 \sinh (3 (e+f x))+864 c d f^3 x^2 \sinh (3 (e+f x))+288 d^2 f^3 x^3 \sinh (3 (e+f x))\right )}{6912 a^3 f^3 (1+\coth (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 3.70, size = 223, normalized size = 0.91
method | result | size |
risch | \(\frac {d^{2} x^{3}}{24 a^{3}}+\frac {d c \,x^{2}}{8 a^{3}}+\frac {c^{2} x}{8 a^{3}}+\frac {c^{3}}{24 a^{3} d}+\frac {3 \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}+2 d^{2} f x +2 c d f +d^{2}\right ) {\mathrm e}^{-2 f x -2 e}}{32 a^{3} f^{3}}-\frac {3 \left (8 d^{2} x^{2} f^{2}+16 c d \,f^{2} x +8 c^{2} f^{2}+4 d^{2} f x +4 c d f +d^{2}\right ) {\mathrm e}^{-4 f x -4 e}}{256 a^{3} f^{3}}+\frac {\left (18 d^{2} x^{2} f^{2}+36 c d \,f^{2} x +18 c^{2} f^{2}+6 d^{2} f x +6 c d f +d^{2}\right ) {\mathrm e}^{-6 f x -6 e}}{864 a^{3} f^{3}}\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 1.13, size = 272, normalized size = 1.11 \begin {gather*} \frac {1}{96} \, c^{2} {\left (\frac {12 \, {\left (f x + e\right )}}{a^{3} f} + \frac {18 \, e^{\left (-2 \, f x - 2 \, e\right )} - 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac {{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} + 108 \, {\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \, {\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 4 \, {\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d e^{\left (-6 \, e\right )}}{576 \, a^{3} f^{2}} + \frac {{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} + 648 \, {\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \, {\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} + 8 \, {\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{2} e^{\left (-6 \, e\right )}}{6912 \, a^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 571 vs.
\(2 (226) = 452\).
time = 0.39, size = 571, normalized size = 2.32 \begin {gather*} \frac {8 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 24 \, {\left (36 \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} + 6 \, c d f + 18 \, {\left (6 \, c d f^{3} + d^{2} f^{2}\right )} x^{2} + d^{2} + 6 \, {\left (18 \, c^{2} f^{3} + 6 \, c d f^{2} + d^{2} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 81 \, {\left (8 \, d^{2} f^{2} x^{2} + 8 \, c^{2} f^{2} + 12 \, c d f + 7 \, d^{2} + 4 \, {\left (4 \, c d f^{2} + 3 \, d^{2} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 3 \, {\left (648 \, d^{2} f^{2} x^{2} + 648 \, c^{2} f^{2} + 540 \, c d f + 8 \, {\left (36 \, d^{2} f^{3} x^{3} - 18 \, c^{2} f^{2} - 6 \, c d f + 18 \, {\left (6 \, c d f^{3} - d^{2} f^{2}\right )} x^{2} - d^{2} + 6 \, {\left (18 \, c^{2} f^{3} - 6 \, c d f^{2} - d^{2} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 243 \, d^{2} + 108 \, {\left (12 \, c d f^{2} + 5 \, d^{2} f\right )} x\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )}{6912 \, {\left (a^{3} f^{3} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3} + 3 \, a^{3} f^{3} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 3 \, a^{3} f^{3} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + a^{3} f^{3} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2443 vs.
\(2 (252) = 504\).
time = 1.37, size = 2443, normalized size = 9.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 315, normalized size = 1.28 \begin {gather*} \frac {{\left (288 \, d^{2} f^{3} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c d f^{3} x^{2} e^{\left (6 \, f x + 6 \, e\right )} + 864 \, c^{2} f^{3} x e^{\left (6 \, f x + 6 \, e\right )} + 1296 \, d^{2} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 648 \, d^{2} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} + 144 \, d^{2} f^{2} x^{2} + 2592 \, c d f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 1296 \, c d f^{2} x e^{\left (2 \, f x + 2 \, e\right )} + 288 \, c d f^{2} x + 1296 \, c^{2} f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 1296 \, d^{2} f x e^{\left (4 \, f x + 4 \, e\right )} - 648 \, c^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 324 \, d^{2} f x e^{\left (2 \, f x + 2 \, e\right )} + 144 \, c^{2} f^{2} + 48 \, d^{2} f x + 1296 \, c d f e^{\left (4 \, f x + 4 \, e\right )} - 324 \, c d f e^{\left (2 \, f x + 2 \, e\right )} + 48 \, c d f + 648 \, d^{2} e^{\left (4 \, f x + 4 \, e\right )} - 81 \, d^{2} e^{\left (2 \, f x + 2 \, e\right )} + 8 \, d^{2}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{6912 \, a^{3} f^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.34, size = 234, normalized size = 0.95 \begin {gather*} {\mathrm {e}}^{-6\,e-6\,f\,x}\,\left (\frac {18\,c^2\,f^2+6\,c\,d\,f+d^2}{864\,a^3\,f^3}+\frac {d^2\,x^2}{48\,a^3\,f}+\frac {d\,x\,\left (d+6\,c\,f\right )}{144\,a^3\,f^2}\right )+{\mathrm {e}}^{-2\,e-2\,f\,x}\,\left (\frac {6\,c^2\,f^2+6\,c\,d\,f+3\,d^2}{32\,a^3\,f^3}+\frac {3\,d^2\,x^2}{16\,a^3\,f}+\frac {3\,d\,x\,\left (d+2\,c\,f\right )}{16\,a^3\,f^2}\right )-{\mathrm {e}}^{-4\,e-4\,f\,x}\,\left (\frac {24\,c^2\,f^2+12\,c\,d\,f+3\,d^2}{256\,a^3\,f^3}+\frac {3\,d^2\,x^2}{32\,a^3\,f}+\frac {3\,d\,x\,\left (d+4\,c\,f\right )}{64\,a^3\,f^2}\right )+\frac {c^2\,x}{8\,a^3}+\frac {d^2\,x^3}{24\,a^3}+\frac {c\,d\,x^2}{8\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________